The interaction between stirring and osmosis. Part 1

When pure solvent is separated from a solution of non-zero concentration C_b by a semi-permeable membrane, permeable to solvent (water) but not to solute, water flows osmotically across the membrane towards the solution. Its velocity J is given by J = PΔC, where P is a constant and ΔC is the concentration difference across the membrane. Because the osmotic flow advects solute away from the membrane, ΔC is usually less than C_b, by a factor γ which depends on the thickness of and flow in a concentration boundary layer. In this paper the layer is analysed on the assumption that the stirring motions in the bulk solution, which counter the osmotic advection, can be represented as two-dimensional stagnation-point flow. The steady-state results are compared with those of the standard physiological model in which the layer has a given thickness δ and the osmotic advection is countered only by diffusion. It turns out that the standard theory, although mechanistically inadequate, accurately predicts the value of γ over a wide range of values of the governing parameter β = PC_bδ/D (where D is the solute diffusivity) if δ is given by \[ \delta = 1.59\bigg(\frac{D}{\nu}\bigg)^{\frac{1}{3}}\bigg(\frac{\nu}{\alpha}\bigg)^{\frac{1}{2}}, \] where ν is the kinematic viscosity of the fluid and α is the stirring parameter. The final approach to the steady state is also analysed, and it is shown to be achieved in a time scale (D/ν)^1/3/αk′ where k′ is a dimensionless number whose dependence on β is computed. Moreover, if β exceeds a certain critical value (≈ 10), the approach to the steady state is not monotonic but takes the form of a damped oscillation (in practice, however, β is unlikely to rise significantly above 1). The theory is extended to the case where the solute concentration is non-zero on both sides of the membrane and in that case it is shown that J is bounded as β → ∞.